Question

Ive been stuck on this forever now if someone could please walk me through it i was appreciate it:

1: Suppose John had a utility function of U=X^2/3Y^1/3 . Derive Johns demand function from his utility function showing all the necessary steps.

i know that the MUx=MUy and first i derive the equation to
2/3X^-1/3 1/3Y^-2/3

then im stuck i don't know what hes asking or how to simplify it do i just solve for X and Y? help please!

Answer 1

I remember how to do this as "one-to-one" mapping problem, which can be solved graphically.
Let me try to solve with calculas and algebra.
CAVEAT EMPTOR - let the buyer beware.

First its MUx/Px=MUy/Py or MUx/MUy = Px/Py.

MUx,MUy are the first derivitive of U.
MUx = (2/3)X^-1/3 Y^1/3
MUy = (1/3)Y^-2/3 X^2/3

Put MUx over MUy (i.e. MUx/MUy) then collapse like terms. That is:
MUx/MUy = 2Y/X = Px/PY
Thus: Px = 2PyY/X

Let Z = total (fixed income). The guy spends all his income on X and Y. So,
Z=PxX + PyY so PyY = Z-PxX substitute this into the equation above. Thus:
Px = 2(Z-PxX)/X = 2Z/X - 2Px soooooo

Px = (2/3)Z/X a nice little demand function

QED

Answer 2

Thank you soo much im a little confused with the simplification so MUx/MUy simplifies to 2y over x ? and for some reason i thought that it would have to end in a demand function of X = something